Free energy and critical temperature in eleven dimensions

TL;DR

This paper calculates the one-loop free energy in eleven-dimensional supergravity with a compactified dimension, explores non-perturbative effects, and investigates the critical temperature for tachyonic states in type IIA string theory at strong coupling.

Contribution

It provides a finite one-loop free energy calculation in M-theory and analyzes the critical temperature for tachyonic states in type IIA string theory at strong coupling.

Findings

Finite one-loop free energy in eleven-dimensional supergravity.

Identification of a tachyonic winding membrane state above a critical temperature.

Critical temperature approaches 0.31 times the inverse Planck length at strong coupling.

Abstract

We compute the one-loop contribution to the free energy in eleven-dimensional supergravity, with the eleventh dimension compactified on a circle of radius $R_{11}$. We find a finite result, which, in a small radius expansion, has the form of the type IIA supergravity free energy plus non-perturbative corrections in the string coupling $g_A$, whose coefficients we determine. We then study type IIA superstring theory at finite temperature in the strong coupling regime by considering M-theory on $R^9\times T^2$, one of the sides of the torus being the euclidean time direction, where fermions obey antiperiodic boundary conditions. We find that a certain winding membrane state becomes tachyonic above some critical temperature, which depends on $g_A$. At weak coupling, it coincides with the Hagedorn temperature, at large coupling it becomes $T_{\rm cr} \cong 0.31 l_P^ {-1} $ (so it is very small in string units).